If a subgroup $H\subset S_n$ is generated by a single $m$-cycle, i.e there is some $c_m=(a_1~a_2~...~a_m)$ such that $H=\left<c_m\right>$. Using orbit-stablizer, $$\left|\operatorname{N}_{S_n}(\left<c_m\right>)\right| = \frac{\left|S_n\right|}{\left|\{ \left<\sigma c_m\sigma^{-1}\right>:\sigma\in S_n\}\right|}$$
The conjugation $\sigma c_m\sigma^{-1}$ is an $m$-cycle and every $m$-cycle in $S_n$ appears in this form for some $\sigma\in S_n$. Note that there are $\phi(m)$ number of $m$-cycles in each $\left<\sigma c_m\sigma^{-1}\right>$, while there are $$\frac{n!}{(n-m)!m}$$ many $m$-cycles in $S_n$. Then $$\left|\{ \left<\sigma c_m\sigma^{-1}\right>:\sigma\in S_n\}\right|=\frac{n!}{(n-m)!m\phi(m)}$$ and $$\left|\operatorname{N}_{S_n}(\left< c_m \right>) \right|=(n-m)!\phi(m)m.$$
In general for an element $\sigma\in S_n$ such that there are $d_1$ copies of subcycles of length $m_1$,and $d_2$ copies of subcycles of length $m_2$,...,and $d_k$ copies of subcycles of length $m_k$(for example $(12)(34)(567)$ has $2$ copies of subcycle of length $2$, and $1$ copy of subcycle of length 3), orbit-stablizer in a similar way gives $$\left|\operatorname{N}_{S_n}\left(\left<\sigma\right>\right)\right|=\left(n-s\right)!\phi\left(\operatorname{lcm}(m_1,...,m_k)\right)\prod_{i=1}^k\left(m_i^{d_i}\right)\prod_{i=1}^k\left(d_i!\right)$$ where $$s=\sum_{i=1}^km_i^{d_i}.$$ Just note that the $\prod (d_i!)$ is the overcount for the orbit as disjoint cycles of the same length may commute.
The case where $H$ is generated by a single permutation is handled above completely. However, the several generators case confuses me. If $\sigma_1,\sigma_2\in S_n$ are two permutations, for example, how can one express $\left|\operatorname{N}(\left<\sigma_1,\sigma_2\right>)\right|$ in terms of $\left|\operatorname{N}(\left<\sigma_1\right>)\right|$ and $\left|\operatorname{N}(\left<\sigma_2\right>)\right|$?($\sigma_1$ and $\sigma_2$ are not necessarily disjoint.)
In general how can one express $\left|\operatorname{N}(\left<\sigma_1,\sigma_2,...,\sigma_r\right>)\right|$ in terms of $\left|\operatorname{N}(\left<\sigma_1\right>)\right|$,$\left|\operatorname{N}(\left<\sigma_2\right>)\right|$,...,$\left|\operatorname{N}(\left<\sigma_r\right>)\right|$?